Approximation of variable-radius offset curves and its application to Bézier brush-stroke design

We present an algorithm to approximate the variable radius ooset curves by cubic B ezier curves. The ooset curve is approximated by a cubic B ezier curve which interpolates the positions and derivatives of the exact ooset curve at both end points. Thus, it approximates the exact ooset curve very closely near the curve end points, but not necessarily in the middle of the curve. Given a xed base curve, by changing the ooset radius and its derivative at an end point, one can easily control the ooset curve shape near the end point. To better control the ooset curve shape in the middle of the curve, we use two global shape parameters (bias and tension) of the ooset curve (see 16]). A variety of variable radius ooset curves are easily generated by using six shape control parameters (the two oosets and their derivatives at both end points, and the bias and tension parameters). Each bristle of a B ezier brush stroke is represented by a variable radius ooset curve. Even with simple linear interpolations of the six shape control parameters, the bristles are generated as cubic interpolations of the two boundary curves. The non-linear interpolability gives a great exibility in modeling exible brush stroke shapes. As the shape control parameters are interpolated , the associated geometric meanings are also interpolated, which makes the designed brush stroke shapes look more natural. This is an improvement over the two previous methods 4, 19] which is based on linear interpolations of the two boundary curves. By applying the same interpolation scheme to the time span as well as to the brush stroke width, it becomes relatively easy to generate exible animated motions of brush strokes.